Tagged Questions

A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that |xy| ≤ |x||y|. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use (c-star-algebras) ...

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Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
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Invertibility of elements in a Banach algebra

Let $X=L^1\cap L^2$, and $\hat{X}$ be the Banach algebra of the image under Fourier transform of $X$. Then do the unital extension $1\dot{+}\hat{X}$ of $X$ by adding a constant function with the norm ...
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Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
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Examples of a Banach space with an algebra structure having only left continuity

There is a theorem (see for example, Rudin's Functional Analysis, theorem 10.2 ) that if $A$ is a Banach space with an algebra structure, such that both left and right multiplication are continuous, ...
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trace norm and tensor product

Let $(M_n (\mathbb{C}), n\|.\|)$ , $(M_n (\mathbb{C}), n\|.\|)$ and $(M_{nm} (\mathbb{C}), nm\|.\|)$ be three Banach algebras. where $$\|A\| = \mathrm{tr}\sqrt{(A^* A)}.$$ What is the norm of $\phi$ ...
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Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V):=B(V,V)$ are linear bounded endomorphisms with operator norm? If so, ...
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Unital amenable Banach algebras which is a proper two sided ideal in its second dual

I need some examples of "unital amenable Banach algebras which is a proper two sided ideal in its second dual".
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Definition of $K_1(A)$ for a Banach algebra $A$

When defining $K_1(A)$ for a Banach algebra $A$, one may consider $\bigcup_{n\in\mathbb{N}}\{x\in GL_n(A^+):x\equiv I_n\mod M_n(A)\}$ and take the quotient by the component containing the identity, or ...
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Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D:A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense ...
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Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
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Density of operators

I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by $\Phi(T)=T^{**}$;...
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About the Banach algebra $\ell^{\infty} (K( \ell^{1}))$

If $P_{n}: \ell^{1} \rightarrow \ell^{1}$ is the projection onto the first n coordinates, then it's well known that $P_{n} K(\ell^{1}) P_{n}$ is isomorphic to $B(\ell^{1}_{n})$, the Banach space of ...
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Toeplitz operators on $\ell_p$ modulo compact operators

There is the well-known extension of $C^*$-algebras $0\rightarrow K\rightarrow\mathcal{T}\rightarrow C(S^1)\rightarrow 0$ where $\mathcal{T}$ is the Toeplitz algebra (generated by the unilateral shift ...
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Question on maximal left ideals.

Suppose I have a unital Banach algebra $A$ and a maximal left ideal $L$ and an element $a$ such that $La\subset{}L$. I want to show that there exists (uniqueness is easy) some complex $\lambda$ such ...
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Operator norm of matrix of scalars regarded as matrix with entries in the unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+=\{(a,z):a\in A,z\in\mathbb{C}\}$ with product $(a,z)(b,w)=(ab+zb+wa,za)$ and norm $||(a,z)||=||a||+|z|$. Equip $M_n(A^+)$ with the operator norm by ...
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Approximation by elements in intersection of two Banach subalgebras

Let $A$ be a Banach algebra, and let $A_1,A_2$ be Banach subalgebras of $A$. Suppose that there exists $c>0$ such that whenever $a_i\in A_i$ ($i=1,2$) and $||a_1-a_2||<\varepsilon$, then there ...
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Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
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Proof of theorems in the field of banach-and $c^*$-algebras in a categorial language

At the moment I'm studying the basics in the theory of banach- and $c^*$-algebras. There are many results in the theory of $c^*$-algebra which you first prove in the unital case and then in the ...
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positive elements in $L(H)$

At the moment I'm studying positive elements in $C^*$-algebras. Let $H$ be a complex Hilbert space, $L(H)$ the linear bounded operators $H\to H$ and $T^*=T$. Then: \sigma(T)\subseteq [0,\|T\| ]\; \...
For $C^*$-algebra tensor products, one talks about the min and max tensor norms, and they agree when one of the $C^*$-algebras is nuclear. For general Banach algebras, what is the analog of nuclearity?...
Are there examples of Banach Algebra's for which the Gelfand transform is 1-1, (ie: the intersection of all maximal ideals of the algebra is $\{0\}$) but not onto? Context Page 96 of Kaniuth's "...